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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rossspw | Structured version Visualization version GIF version |
Description: A ring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.) |
Ref | Expression |
---|---|
isros.1 | ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))} |
Ref | Expression |
---|---|
rossspw | ⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isros.1 | . . . 4 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))} | |
2 | 1 | isros 33061 | . . 3 ⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆))) |
3 | 2 | simp1bi 1145 | . 2 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝒫 𝒫 𝑂) |
4 | 3 | elpwid 4606 | 1 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 ∖ cdif 3942 ∪ cun 3943 ⊆ wss 3945 ∅c0 4319 𝒫 cpw 4597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pw 4599 |
This theorem is referenced by: rossros 33073 |
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